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Double Schubert polynomials and degeneracy loci for the classical groups

Andrew Kresch, Harry Tamvakis (2002)

Annales de l’institut Fourier

We propose a theory of double Schubert polynomials $P_{w} (X, Y)$ for the Lie types $B$ , $C$ , $D$ which naturally extends the family of Lascoux and Schützenberger in type $A$ . These polynomials satisfy positivity, orthogonality and stability properties, and represent the classes of Schubert varieties and degeneracy loci of vector bundles. When $w$ is a maximal Grassmannian element of the Weyl group, $P_{w} (X, Y)$ can be expressed in terms of Schur-type determinants and Pfaffians, in analogy with the type $A$ formula of Kempf and Laksov....

Double-points of compositions.

Johan P. Hansen (1984)

Journal für die reine und angewandte Mathematik

Durchmesser und Mittelpunkte algebraischer Hyperflächen.

Ernst Kunz, Rainer Lux (1996)

Elemente der Mathematik

Effective Nullstellensatz for arbitrary ideals

János Kollár (1999)

Journal of the European Mathematical Society

Let $f_{i}$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_{i}$ such that $\sum g_{i} f_{i} = 1$ . The effective versions of this result bound the degrees of the $g_{i}$ in terms of the degrees of the $f_{j}$ . The aim of this paper is to generalize this to the case when the $f_{i}$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.